We employ the method of multiple scales (two-timing) to
analyse the vortex dynamics of inviscid, incompressible flows that
oscillate in time. Consideration of distinguished limits for Euler's
equation of hydrodynamics shows the existence of two main asymptotic
models for the averaged flows: strong vortex dynamics (SVD)
and weak vortex dynamics (WVD). In SVD the averaged vorticity
is 'frozen' into the averaged velocity field. By contrast, in WVD
the averaged vorticity is 'frozen' into the 'averaged velocity +
drift'. The derivation of the WVD recovers the Craik-Leibovich
equation in a systematic and quite general manner. We show that the
averaged equations and boundary conditions lead to an energy-type
integral, with implications for stability.

Oscillating flows represent an important aspect of
classical fluid dynamics and appear in various applications in
medicine, biophysics, geophysics, engineering, astrophysics and
acoustics. The term "oscillating flow" usually means that the fluid
motion under consideration possesses a dominant frequency $\sigma$,
which can be maintained either by an oscillating boundary condition,
by an oscillating external force, or by self-oscillations of a flow.
All other motions in the oscillating flow are considered as
"slow", with the related
time-scale $T_{{\rm slow}}\gg 1/\sigma$. The scale $T_{{\rm slow}}$
also can be related to a boundary condition, to an external force,
or it may characterise a natural intrinsic motion of the fluid. A
related powerful mathematical approach is the two-timing
method (see, for example, [Nayfeh1973]; [Kevorkian and Cole1996]). In this paper we
use this method, together with the idea of distinguished limits, to
provide an elementary, systematic, and justifiable procedure
following the ideas proposed in [Vladimirov2005], [Yudovich2006],
[Vladimirov2008]; [Vladimirov2010]. The results and contents of this
paper may be summarised as:

1.

The development of a new analytical approach for the description of oscillating-in-time flows. The fluid is assumed to be inviscid and incompressible, with the oscillations introduced via the boundary conditions.

2.

The analysis of distinguished limits for the Euler equation shows the existence of two asymptotic models for the averaged flows: "strong" or "standard" vortex dynamics (SVD), and weak vortex dynamics (WVD), the latter described by the Craik-Leibovich equation (CLE). The CLE was originally derived for the description of the Langmuir circulations generated by surface waves ([Craik1985]; [Leibovich1983]; [Thorpe2004]). The derivation of the CLE demonstrates the remarkable fact that the Reynolds stresses can be expressed solely in terms of the drift velocity.

3.

In SVD the averaged vorticity is frozen into the averaged velocity, whereas in WVD the averaged vorticity is frozen into the averaged velocity $+$ drift velocity. It is important that in WVD the drift velocity has the same order of magnitude as the averaged velocity. Our derivation of the WVD and CLE is technically simpler than previous derivations. The formulation of the problem in its natural generality shows that the area of applicability of the CLE is broader than previously recognised. In particular, we consider flow domains that are three-dimensional and of arbitrary shape; the oscillations are time-periodic, but their spatial structure is arbitrary. We have also derived the averaged boundary conditions that are valid at the average positions of the boundaries.

4.

The slow time-scale is uniquely linked to the magnitude of the prescribed velocity field at the boundary. Naturally, the higher the amplitude of velocity, the shorter the slow time-scale.

5.

The WVD and CLE contain the drift velocity. The drift usually appears as the average velocity of Lagrangian particles (see [Stokes1847]; [Lamb1932]; [Batchelor1967]). In our consideration, drift velocity appears naturally as the result of an Eulerian averaging of the related PDEs without directly addressing the motions of particles.

6.

The CLE leads to an energy-type integral for the averaged flows, which allows us to consider "Arnold-type" results, such as the generalized "isovorticity conditions", the energy variational principle, the first and second variation of energy, and several (nonlinear and/or linear) stability criteria for averaged flows.

We study the motion of a homogeneous inviscid incompressible fluid
in a time-dependent three-dimensional domain $Q(t)$ with oscillating
boundary $\partial Q(t)$ (see Fig. 1), which is prescribed
as

$F^{\dagger}({\boldsymbol{x}}^{\dagger},t^{\dagger})=0.$

(1)

The velocity field ${\boldsymbol{u}}^{\dagger}={\boldsymbol{u}}^{\dagger}({\boldsymbol{x}}^{%
\dagger},t^{\dagger})$ and vorticity ${\boldsymbol{\omega}}^{\dagger}\equiv\nabla^{\dagger}\times{\boldsymbol{u}}^{\dagger}$ satisfy the equations

where ${\boldsymbol{x}}^{\dagger}=(x_{1}^{\dagger},x_{2}^{\dagger},x_{3}^{\dagger})$ are Cartesian coordinates, ${t}^{\dagger}$ is time, daggers denote dimensional variables, and square brackets stand for the commutator of two vector fields $[{\boldsymbol{a}},{\boldsymbol{b}}]\equiv({\boldsymbol{b}}\cdot\nabla){%
\boldsymbol{a}}-({\boldsymbol{a}}\cdot\nabla){\boldsymbol{b}}$. The kinematic boundary condition at $\partial Q$ is

The natural small parameter in our consideration is $1/\sigma$. The
essence of the two-timing method is based on the assumption that the
ratio $T_{\text{slow}}/T_{\text{fast}}=S/\sigma$ also represents a small
parameter. As a result, Eq. (10) contains two
independent small parameters, $\varepsilon_{1}$ and $\varepsilon_{2}$:

Then, in the two-timing method, we make the standard auxiliary (but
technically essential) assumption that the variables $s$ and $\tau$
are (temporarily) considered to be mutually independent.
Its justification can be given a posteriori after solving
(11), rewriting the solution in terms of the original
variable $t$, and estimating the errors/residuals in the original
equation (1), also expressed in terms of $t$
([Yudovich2006]).

Let us temporarily forget about the definitions of $\varepsilon_{1}$ and
$\varepsilon_{2}$ in (11) and treat them as abstract
small parameters. In order to construct a rigorous asymptotic
procedure with $(\varepsilon_{1},\varepsilon_{2})\to(0,0)$ we have to
consider the various paths approaching the origin in the
$(\varepsilon_{1},\varepsilon_{2})$-plane. One may expect that there
are infinitely many different asymptotic solutions to
(11) corresponding to different paths (the usual
sequence of the limits $\varepsilon_{1}\to 0$ and then
$\varepsilon_{2}\to 0$, or with the order reversed, correspond to the
'broken' paths). However, for (11) (as well as for
many other equations) one can find a few exceptional paths, which we
shall call the distinguished limits. The notion of a
distinguished limit is imprecisely defined (see, for example,
[Nayfeh1973]; [Kevorkian and Cole1996]), varying between different books and papers.
We suppose that a distinguished limit is given by a path that allows
us to build a self-consistent asymptotic procedure, leading to a
finite/valid solution in any approximation. No systematic procedure
of finding all possible distinguished paths is known, and so this may be
regarded as still a problem of experimental mathematics.

We have considered in detail a number of different paths
parametrized by

subsequent approximations producing various 'oscillatory' and 'mean'
corrections. This is the case of Strong Vortex Dynamics
(SVD). In contrast, for Eq. (14), the fluid motion is purely
oscillatory in the main approximation,

Hence for the case (14) we consider only a relatively weak
vorticity developing on the background wave motion. This leads to
the Craik-Leibovich equation and to Weak Vortex Dynamics
(WVD). All other cases (12) that we have considered can be
transformed either to one of these two main cases, or else they
produce inconsistent/unsolvable systems of successive
approximations, or else they lead to secular growth in $s$.

Equations (13) or (14) must be complemented by the boundary condition (3), with the same ordering of small parameters. This leads respectively to:

can be split into averaged and purely oscillating parts, $f({\boldsymbol{x}},s,\tau)=\overline{f}({\boldsymbol{x}},s)+\widetilde{f}({%
\boldsymbol{x}},s,\tau)$; the tilde-functions (or purely oscillating functions) are such that $\langle\widetilde{f}\,\rangle=0$ and the bar-functions are $\tau$-independent. Furthermore, we introduce the tilde-integration which keeps the result in the tilde-class:

in the form of the regular series (15) . We restrict the class of possible solutions by imposing (17) .

The equations for successive approximations show that the zeroth order approximation of (22) is $\widetilde{{\boldsymbol{\omega}}}_{0\tau}=0$; its unique solution (within the tilde-class) is $\widetilde{{\boldsymbol{\omega}}}_{0}\equiv 0$. Together with (17) it shows that the full vorticity vanishes,

${\boldsymbol{\omega}}_{0}\equiv 0,$

(23)

which means that the velocity field at leading order is purely oscillatory and potential. Then, similarly, the equation of the first order approximation of (22) yields $\widetilde{{\boldsymbol{\omega}}}_{1\tau}=0$. Its unique solution (within the tilde-class) is $\widetilde{{\boldsymbol{\omega}}}_{1}\equiv 0$, while the mean value $\overline{{\boldsymbol{\omega}}}_{1}$ remains undetermined. We write this symbolically as

It can be seen that if $\widetilde{{\boldsymbol{u}}}_{0}$ is solenoidal then the drift velocity $\overline{{\boldsymbol{V}}}_{0}$ is also solenoidal, i.e. $\nabla\cdot\overline{{\boldsymbol{V}}}_{0}=0$.

After dropping subscripts and bars in $\overline{{\boldsymbol{u}}}_{1}$ and
$\overline{{\boldsymbol{\omega}}}_{1}$, Eq. (29) can be used as
the WVD model for the evolution of the averaged vorticity:

which shows that the averaged vorticity is frozen into the 'velocity
$+$ drift'. This result is known as the Craik-Leibovich
equation (CLE) (see, for example, [Craik1985]). The derivation
of the CLE here is much simpler technically than previous
derivations, and minimises the number of assumptions needed (e.g. those on the flow geometry). We should emphasize that the drift
velocity here is not considered to be small; it is of the same order
of magnitude as the Eulerian averaged velocity.
Equation ((31) may be integrated (in space) as

and (33) gives the is given for both Eqs. (35) and (40), the citation of equations has been changed in text. Kindly check and confirm the citation of Eqs. (35) and (40) throughout the article. averaged 'no-leak' condition:

The effective boundary $\partial Q_{0}$ for this averaged flow is
given by the equation $\overline{F}_{0}({\boldsymbol{x}},s)=0$; it means that the
boundary conditions are prescribed not at the real boundary, but at
its averaged position. Equations (31) ,
(32) and (33) or
(37) form the closed model describing the averaged WVD
flow.

The drift velocity $\overline{{\boldsymbol{V}}}_{0}$ is to be calculated from
(30)$\Lsh$,
where $\widetilde{{\boldsymbol{u}}}_{0}$ represents the
solution of the previous approximation, $\widetilde{{\boldsymbol{u}}}_{0\tau}=-\nabla\widetilde{p}_{0}$ and $\nabla\cdot\widetilde{{\boldsymbol{u}}}_{0}=0$,
together with the boundary condition
$\widetilde{F}_{1\tau}+\widetilde{{\boldsymbol{u}}}_{0}\cdot\nabla\overline{F}_%
{0}$
at $\overline{F}_{0}=0$.

It is possible to find alternative scalings for the slow time scale
$s$, while respecting the constraints given by the distinguished
limits (13) , (14) . The slow time is defined in such a
way that intervals of order one in $s$ correspond to changes of
order one in the physical fields. In the SVD the mean velocity
(16) is $O(1)$, and so we must have $s=t$. Physically,
this means that in order to transport an admixture a dimensionless
distance of order one, we need a dimensional time of order one. In
the WVD the mean velocity (17) is $O(\delta)$. Then advection with $\overline{{\boldsymbol{u}}}_{0}=O(\delta)$ requires the slow
time-scale $s=\delta\,t$ (for $s=1$ the interval of the original
'physical' time is $1/\delta$).

The new formal small parameter $\delta$, introduced to describe the
distinguished limit, can be related to the fast time scale
$1/\sigma$ in a number of different ways. It is instructive to
rewrite Eq. (22) as

with constants $\alpha$ and $\beta$. In order to make
(22) coincide with (38) we require

$\beta=(1-\alpha)/2,\quad\delta=1/\sigma^{(\alpha+1)/2}.$

(39)

Equations (38) , (39) can be
interpreted in the following way: the related slow time-scale is
$s=t/\sigma^{\alpha}$ (where $\alpha>-1$, which means that $s$ is a
'slow' variable in comparison with $\tau$) and the velocity is
$\sigma^{\beta}{\boldsymbol{u}}$, not ${\boldsymbol{u}}$.

This transformation allows one to vary the slow time-scale.
Consider, for example, the case when [instead of (20) ]
the boundary is prescribed as

which can appear in many practical applications. In this case the
slow time-scale $s=t$ is prescribed by the boundary condition; in
fact we have $\alpha=0$ in (39) (and so for WVD we
have the same time scales as for SVD) namely

$\tau=\sigma t,\qquad s=t.$

(41)

However, if this is to be the correct scaling in the WVD then we
must have $\beta=1/2$, so that the velocity at the boundary is
$O(\sqrt{\sigma})$, not $O(1)$; also, from (39) , the
small parameter of the decomposition should be chosen as
$\delta=1/\sqrt{\sigma}$. Another interesting possibility corresponds
to $\alpha=\beta=1/3$. In this case, $s=t/\sqrt[3]{\sigma}$,
$\delta=\sigma^{-2/3}$ and the velocity is $\sqrt[3]{\sigma}{\boldsymbol{u}}$.
Although such an asymptotic scaling may look exotic, it would be
required if the particular slow time-scale $s=t/\sqrt[3]{\sigma}$ were
prescribed by the boundary conditions. The original case
(11) , which corresponds to an $O(1)$ velocity,
corresponds to $\alpha=1$ and $\beta=0$. The general tendency is
physically natural: to shorten the slow time-scale (decreasing
$\alpha$), one needs to increase the amplitude of the boundary
oscillations (increasing $\beta$) (see Eq. (39) ).

In order to connect the above model equations
(31) with classical areas of fluid dynamics, let
us show that for a plane surface wave $\overline{{\boldsymbol{V}}}_{0}$, from
(30)$\Lsh$, gives the classical Stokes drift and that this
then leads to an understanding of the nature of Langmuir vortices.

The dimensional solution for a plane potential travelling gravity wave is

which agrees with the classical expression for the drift velocity
([Lamb1932]; [Debnath1994]; [Batchelor1967]). To obtain
(42) one should take into account that the
transformation to the physical formula for drift includes a move
from the slow time $s=t/\sigma$ to the physical time $t$.

The structure of the CLE and WVD can be seen as a
relatively passive alteration of the original Euler equations, since
we still have frozen-in vorticity dynamics. However, the additional
terms (which contain the drift velocity) make a qualitative change
to the properties of the solutions. One example of such a new
property is related to the Langmuir circulations (see
Fig. 2). In order to illustrate such a qualitative
change, let us consider the class of translationally invariant
averaged flows. Let the zeroth approximation (23) take
the form of a plane potential travelling gravity wave with the drift
velocity (30). Let Cartesian coordinates $(x,y,z)$
be such that $\overline{{\boldsymbol{V}}}_{0}=(U,0,0)$, $U=e^{2z}$,
$\overline{{\boldsymbol{u}}}_{1}=(u,v,w)$, where all components are $x$-independent
(translationally-invariant).

Figure 2: Langmuir circulations are mathematically similar to the Rayleigh-Taylor instability of inversely stratified fluid

where $\rho\equiv u$, $\Phi\equiv U=e^{2z}$ and $\overline{P}$ is a
new modified pressure. One can see that Eq. (43) are
mathematically equivalent to the system of equations for an
incompressible stratified fluid in the Boussinesq approximation. The
effective 'gravity field' ${\boldsymbol{g}}=-\nabla\Phi=(0,0,-2e^{2z})$ is
non-homogeneous, which makes the analogy with the 'standard'
stratified fluid incomplete. Nevertheless, longitudinal vortices
should naturally appear in (43) as a Rayleigh-Taylor
type instability of an inversely stratified equilibrium
corresponding to $(u,v,w)=(u(z),0,0)$ with any increasing function
$u(z)\equiv\rho(z)$ (see Fig. 2).

According to (32) , vorticity is frozen into
${\boldsymbol{u}}+\overline{{\boldsymbol{V}}}_{0}$. We may then use a slightly modified Arnold
isovorticity condition ([Arnold and Khesin1999]) in its differential form,

where ${\boldsymbol{u}}({\boldsymbol{x}},\theta)$ is the unknown function,
${\boldsymbol{f}}={\boldsymbol{f}}({\boldsymbol{x}},\theta)$ is an arbitrary given solenoidal function,
$\theta$ is a scalar parameter along an isovortical trajectory, and
subscript $\theta$ denotes a partial derivative. The function
$\alpha({\boldsymbol{x}},\theta)$ is to be determined from the condition $\nabla\cdot{\boldsymbol{u}}=0$. The initial data at $\theta=0$ for ${\boldsymbol{u}}({\boldsymbol{x}},\theta)$
in (46) corresponds to a steady flow

where ${\boldsymbol{U}}({\boldsymbol{x}})$ and ${\boldsymbol{\Omega}}({\boldsymbol{x}})$ represent the steady solutions
($\partial/\partial s=0$) of (31) and
(32) with the no-leak boundary conditions
(35) .

Differentiation of $E$ with respect to $\theta$ produces the first variation

which vanishes for any solenoidal function ${\boldsymbol{f}}$ by virtue of the
equations of motion and the boundary conditions for steady flow.
This equality gives us the variational principle: any steady flow
represents a stationary point on the isovortical sheet. The only
difference from Arnold's classical result is the boundary conditions
in the definition of the isovorticity sheet (46) .

where ${\boldsymbol{W}}\equiv{\boldsymbol{U}}+\overline{{\boldsymbol{V}}}_{0}$. This is analogous to
Arnold's result; expression (49) shows that the
stationary point of the energy functional in the three-dimensional
case always represents a saddle point.

However, the stability conditions can be obtained for steady plane
flows in the case when a stream function $\Psi(x_{1},x_{2})$ for the
combined velocity ${\boldsymbol{W}}(x_{1},x_{2})$ can be introduced as
$W_{1}=\partial\Psi/\partial x_{2}$, $W_{2}=-\partial\Psi/\partial x_{1}$.
For the plane flow the second variation (49) , combined
with the standard Casimir integral containing an arbitrary function
of vorticity ([Arnold and Khesin1999]) takes the form

where $\omega({\boldsymbol{x}},\theta)$ and $\Omega({\boldsymbol{x}})$ are the $x_{3}$-components of the full vorticity and the steady vorticity at $\theta=0$, and the functional dependence $\Psi=\Psi(\Omega)$ characterises the plane steady flow under consideration. Then, similarly to the Arnold cases, the inequalities with two positive (or two negative) constants $C^{-}$, $C^{+}$ satisfying

$C^{-}<-\frac{d\Psi}{d\Omega}<C^{+}$

(51)

give both sufficient linear and nonlinear stability conditions for
the positively (and negatively) defined energy-casimir functional.
One should take into account that these stability conditions
determine stability with respect to arbitrary perturbations, not
only isovortical ones. However, this part of the analysis is similar
to Arnold's well-known results of 1966 (see [Arnold and Khesin1999]) and is
not presented here.

Finally, we note that for plane WVD flows we have derived sufficient stability conditions that differ from the classical ones only by replacing the streamfunction for the velocity field ${\boldsymbol{U}}$ by the streamfunction for the combined velocity ${\boldsymbol{W}}$. Therefore an important conclusion for the stability of the plane WVD flows is that virtually any plane steady flow can be made stable by the choice of the 'proper' field of drift velocity.

Our main achievement in this paper is a significant simplification of the derivation of the Craik-Leibovich equation (CLE). Its most known derivation (
[Craik1985]
) is performed with the use of the Generalized Lagrangian Mean theory (GLM) (see
[Andrews and McIntyre1978]
;
[Craik1985];
[Buhler2009]) and further theoretical studies are often performed in GLM terms (
[Holm1996]). In contrast, here we introduce the CLE in its natural simplicity and generality and use only the standard Eulerian description,
making our derivation much more accessible.

Examples of oscillating flows are not restricted by a deformed
domain as in Fig.
1
; they can be extended to many oscillating flows that appear in practical applications, such asoscillating or rotating rigid bodies, moving pistons and acoustics
([Vladimirov and Ilin2013]); some of these cases are
illustrated in Fig.
3.

Figure 3: Various
practically important examples of weak vortex dynamics in a fluid
that undergoes externally imposed
oscillations

The two small parameters that we have used represent two ratios of
three time-scales. It should be noted that in the derivation of CLE
we have not used the amplitude as a small parameter. The main field
of velocity oscillations $\widetilde{{\boldsymbol{u}}}_{0}$ is of dominant order and so is not small. Nevertheless, the oscillatory
spatial amplitude of material particles and the related spatial
amplitude of the deformation of the boundary $\partial Q(t)$ are
both small.

It is instructive to derive the averaged equations of the second
approximation (while the CLE
(29) ) appears in the
first approximation). This appears as the linearized
equation at the first approximation and contains an additional
'force term' depending on the previous approximations. It means that
some additional instabilities are possible, beyond classical
instabilities of the linearized problem. This raises interesting
questions about the meaning of linearization and its non-uniqueness.

It is remarkable that the same CLE
(29)
describes
the WVD in the case of acoustics (see [Vladimirov and Ilin2013]), when
$\widetilde{{\boldsymbol{u}}}_{0}$ represents a given acoustic wave that satisfies
the wave equations and cannot be solenoidal. An important
qualitative addition of the 'acoustical CLE' is that the drift
velocity can be an arbitrary solenoidal function. It gives greater
general significance to studies of the CLE, since an arbitrary
solenoidal function $\overline{{\boldsymbol{V}}}_{0}$, Eq.
(30)
now has a practical meaning.

A viscous term can be straightforwardly added to the right hand side
of the CLE (see
[Craik1985]). Our derivation
(22)
- (30)$\Lsh$ shows that in order to accommodate such an addition the dimensionless
viscosity (or the inverse Reynolds number) should be of order
$\delta^{3}$.

The same analysis as above is valid for stratified fluids in the
Boussinesq approximation where the generalization of the CLE is
straightforward (see
[Craik1985]). At the same time, the analogy with stratification
(43)
discussed earlier leads to Richardson-type stability criteria even in the case of the CLE for homogeneous fluid.

The CLE is Hamiltonian as is immediately clear from its form. This
question was considered by [Holm1996] for the CLE in the GLM form
introduced by [Andrews and McIntyre1978],
[Craik1985],
[Buhler2009], which
is somewhat different from ours. However, the investigation of the
Hamiltonian structure of our equations is beyond the scope of this
paper. Similarly, it would be of interest to study the possibility
of a finite-time vorticity singularity for the CLE.

A generalization of the CLE and WVD has been obtained for MHD by
[Vladimirov2013].
In this case the drift velocity appears in both the equation for the advection of vorticity and the equation for the
advection of magnetic field. Similar results restricted to
kinematic MHD have also been obtained in [Vladimirov2010] and [Herreman and Lesaffre2011]. There remains the challenge of developing a
self-consistent theory of the full MHD equations
([Moffatt1978]), which may be viewed as a complementary approach
to that of [Courvoisier et al.2010].